Crossword Genius

Cotton on cut that’s bandaged in hospital (6)

Ross

I believe the answer is:

clinic

I'm a little stuck... Click here to teach me more about this clue!

'hospital' is the definition.
(I've seen this before)

'cotton on cut that's bandaged in' is the wordplay.
I cannot quite see how this works, but
'in' is found in the answer.
This may be the basis of the clue (or it may be nonsense).

Can you help me to learn more?

(Other definitions for clinic that I've seen before include "Healthcare facility" , "Private hospital" , "Old bedridden person" , "Surgery" , "Doctor's session" .)

I've seen this clue in The Guardian.
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